Answer:
a(i) 0
a(ii) π
b) [0, 4)
Step-by-step explanation:
a(i) ∑ₙ₌₁°° aₙ = π
The series converges, which means lim(n→∞) aₙ = 0.
a(ii) sₙ is the partial sum, so lim(n→∞) sₙ = π.
b) Use ratio test:
lim(n→∞)│aₙ₊₁ / aₙ│< 1
lim(n→∞)│[(3x−6)ⁿ⁺¹ / ((n+1)6ⁿ⁺¹)] / [(3x−6)ⁿ / (n6ⁿ)]│< 1
lim(n→∞)│[(3x−6)ⁿ⁺¹ / ((n+1)6ⁿ⁺¹)] × [(n6ⁿ) / (3x−6)ⁿ]│< 1
lim(n→∞)│(3x−6) n / (6(n+1))│< 1
│(3x−6) / 6│< 1
│3x−6│< 6
-6 < 3x − 6 < 6
0 < 3x < 12
0 < x < 4
Check the endpoints.
If x = 0, ∑ₙ₌₁°° (3(0)−6)ⁿ / (n6ⁿ) = ∑ₙ₌₁°° (−1)ⁿ / n, which converges.
If x = 4, ∑ₙ₌₁°° (3(4)−6)ⁿ / (n6ⁿ) = ∑ₙ₌₁°° 1 / n, which diverges.
So the interval of convergence is [0, 4).
